数论

We study the class of polynomials that map a local field (i.e., the completion of a number field at a non-Archimedean place) into the subset of its $p$-th powers, where $p$ is the residue characteristic of the field in question. We present…

We study the extensibility of $D(4)$-pairs $\{a,b\}$, where $b = ka$ and $k \in \{7,8,10,11,12,13\}$. Firstly, we show that it can be extended to a $D(4)$-triple with an element c, which is a member of a family of positive integers…

In this article, we establish a new linear independence criterion for the values of certain {\it Lauricella hypergeometric series} $F_D$ with rational parameters, in both the complex and $p$-adic settings, over an algebraic number field.…

Baba and Granath generalize Elkies' theorem on infinitude of supersingular primes for elliptic curves to abelian surfaces with quaternionic multiplication of discriminant $6$, whose field of moduli is $\mathbb{Q}$ and which is a Jacobian in…

Let $k$ be a number field. We investigate the Mordell-Weil ranks of Jacobian varieties $J_C$ associated with algebraic curves $C$ of genus $g \geq 1$ defined by affine equations of the form $y^s=x(ax^r+b)$, where $a, b \in k$ ($ab \neq 0$),…

We establish power-saving estimates for general bilinear forms with Kloosterman sums modulo arbitrary q, including when both variables are shorter than the Polya-Vinogradov range. As an application, we obtain power-saving asymptotics for…

The Erdos-Straus conjecture (ESC) concerns the representation of the fraction 4/P, where P is a prime number, as a sum of three positive unit fractions. The focus here is on the case when P is congruent to 1 modulo 4. Two constructive…

We obtain polylogarithmic bounds in the polynomial Szemer\'{e}di theorem when the polynomials have distinct degrees and zero constant terms. Specifically, let $P_1, \dots, P_m \in \mathbb Z[y]$ be polynomials with distinct degrees, each…

We prove a novel zeta regularized product formula concerning regularization of trigonometric products over non-trivial zeros of the Riemann zeta function. Furthermore, we calculate the discrepancies of such regularized products. In special…

Let $n \in \mathbb{Z}_{\geqslant 2}$. By $P(n)$ we denote the set of all prime divisors of the integers in the sequence $n, n^2-1, (n^2-1)^2-1, \dots$. We ask whether the set $P(n)$ determines $n$ uniquely under the assumption that $n \neq…

Let $n,m$ be positive integers and $c \in \mathbb{Z}_n$, where $\mathbb{Z}_n$ is the ring of integers modulo $n$. We almost complete providing the answer to the following problem, partially solved by N. Alon. Does any infinite sequence over…

For any constant $C_0>0$, we construct a set $A \subset {\mathbb N}$ such that one has $$ \sum_{n \in A: n \leq x} \frac{1}{n} = \exp\left(\left(\frac{C_0}{2}+o(1)\right) (\log\log x)^{1/2} \log\log\log x \right)$$ and $$ \sum_{n,m \in A:…

For a profinite group $G$ and a rigid analytic space $X$, we study when an $\mathcal O_X(X)$-linear representation $V$ of $G$ admits a lattice, i.e. an $\mathcal O_{\mathcal X(\mathcal X)}$-linear model for a suitable formal model $\mathcal…

In 2015, Bachmann \cite{Ba3} conjectured that the~$\Q$-vector space~$\Zq$ of (formal)~$q$-analogues of Multiple Zeta Values (\qmzv s) is spanned by a very particular set compared to known spanning sets. In this work, we prove that this…

We prove a substantial part of conjectures of Mazur and Tate that refine the conjecture of Birch and Swinnerton-Dyer. Our approach, which also leads to some results even finer than the predictions of Mazur and Tate, is via the `rank-zero…

In this paper, it is proved that, for $\gamma\in(\frac{317}{320},1)$, every sufficiently large odd integer can be written as the sum of nine cubes of primes, each of which is of the form $[n^{1/\gamma}]$. This result constitutes an…

Vinogradov's three primes theorem indicates that, for every sufficiently large odd integer $N$, the equation $N=p_1+p_2+p_3$ is solvable in prime variables $p_1,p_2,p_3$. In this paper, it is proved that Vinogradov's three primes theorem…

Let $-1/2<a<0$ be a fixed real number and \begin{equation*} \Delta_{a}(x)=\sideset{}{'}\sum_{n\leq x} \sigma_a(n)-\zeta(1-a)x-\frac{\zeta(1+a)}{1+a}x^{1+a}+\frac{1}{2}\zeta(-a). \end{equation*} In this paper, we investigate the…

Let $f \in \mathbb Q[x]$ be a square-free polynomial of degree at least $3$, $m_i$, $i=1,2,3$, odd positive integers, and $a_i$, $i=1,2,3$, non-zero rational numbers. We show the existence of a rational function…

Suppose that $a$ and $b$ are positive integers subject to $(a,b)=1$. For $n\in\mathbb{Z}^+$, denote by $\tau_{a,b}(n;\ell_1,M_1,l_2,M_2)$ the asymmetric two--dimensional divisor function with congruence conditions, i.e., \begin{equation*}…